What is the integral of x^2sqrt(x^3+21) ?

The integral of x^2sqrt(x^3+21) is 2/9 (x^3+21)^{3/2}+C

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integral of x^2sqrt(x^3+21)

\int x^{2} x^{3} + 21 d x

\frac{2}{9} (x^{3} + 21)^{\frac{3}{2}} + C

Solution steps

\int x^{2} x^{3} + 21 d x

Apply u-substitution

: \int \frac{u}{3} d u

= \int \frac{u}{3} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{3} \cdot \int u d u

Apply the Power Rule

: \frac{2}{3} u^{\frac{3}{2}}

= \frac{1}{3} \cdot \frac{2}{3} u^{\frac{3}{2}}

Substitute back

u = x^{3} + 21

= \frac{1}{3} \cdot \frac{2}{3} (x^{3} + 21)^{\frac{3}{2}}

Simplify

\frac{1}{3} \cdot \frac{2}{3} (x^{3} + 21)^{\frac{3}{2}} : \frac{2}{9} (x^{3} + 21)^{\frac{3}{2}}

= \frac{2}{9} (x^{3} + 21)^{\frac{3}{2}}

Add a constant to the solution

= \frac{2}{9} (x^{3} + 21)^{\frac{3}{2}} + C

\int + \frac{x ^{2}}{( 225 + x ^{2} ) ^{2}} d x

x \to 0 lim (\frac{( 3 ^{x} - 4 ^{x} )}{x})

e^{9 x}

\frac{1}{3} sin (3 x)

12 x^{3} - 3 y^{2} x^{4} + 1 y^{^{'}} = 0, y (0) = 2

What is the integral of x^2sqrt(x^3+21) ?
The integral of x^2sqrt(x^3+21) is 2/9 (x^3+21)^{3/2}+C